Trivializing group actions on braided crossed tensor categories and graded braided tensor categories

Abstract

For an abelian group 𝐴, we study a close connection between braided 𝐴-crossed tensor categories with a trivialization of the 𝐴-action and 𝐴-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action 𝑇 on a tensor category 𝒞 is given by an element 𝑂(𝑇) ∈ 𝐻²(𝐺, Aut_⊗(Id_𝒞)). In the case that 𝑂(𝑇) = 0, the set of obstructions forms a torsor over Hom(𝐺, Aut_⊗(Id_𝒞)), where Aut_⊗(Id_𝒞) is the abelian group of tensor natural automorphisms of the identity.

The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided 𝐴-crossed tensor categories developed Etingof et al., allows us to provide a method for the construction of faithfully 𝐴-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided 𝐴-crossed tensor category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided ℤ/2ℤ-crossed structures over Tambara–Yamagami fusion categories and, consequently, a conceptual interpretation of the results by Siehler about the classification of braidings over Tambara–Yamagami categories.